L ∞-Algebra Connections and Applications to String- and Chern-Simons n-Transport

Abstract

We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their parallel transport and quantization.It is known that over a D-brane the Kalb-Ramond background field of the string restricts to a 2bundle with connection (a gerbe) which can be seen as the obstruction to lifting the P U (H)-bundle on the D-brane to a U (H)-bundle. We discuss how this phenomenon generalizes from the ordinary central extension U (1) → U (H) → P U (H) to higher categorical central extensions, like the String-extension BU (1) → String(G) → G. Here the obstruction to the lift is a 3-bundle with connection (a 2-gerbe): the Chern-Simons 3-bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a String-structure. We discuss how to describe this obstruction problem in terms of Lie n-algebras and their corresponding categorified Cartan-Ehresmann connections. Generalizations even beyond String-extensions are then straightforward. For G = Spin(n) the next step is "Fivebrane structures" whose existence is obstructed by certain generalized Chern-Simons 7-bundles classified by the second Pontrjagin class.

An invitation to higher gauge theory

“…There is a lot of sophisticated mathematics involved here, but ultimately much of it should arise from the way string 2-groups are involved in the parallel transport of strings! The work of Sati et al [83] provides good evidence for this, as does the work of Waldorf [92].…”

“…But more recently, Sati et al [82,83] have developed a lot of higher gauge theory with the help of L ∞ -algebras. Together with the insights of Aschieri et al [1,2], their work makes it clear that superstring theory, supergravity and even the mysterious 'M-theory' have strong ties to higher gauge theory.…”

“…Higher gauge theory provides new insights into the geometry of these theories. In particular, the work of Castellani et al [35] can be seen as implicitly making extensive use of Lie n-superalgebras-but this only became clear later, through the work of Sati et al [83].…”

“…This should not be surprising, since while these rival approaches to quantum gravity disagree about almost everything, they both agree that point particles are not enough: we need higher-dimensional extended objects to build a theory sufficiently rich to describe the quantum geometry of spacetime. Indeed, many existing ideas from string theory and supergravity have recently been clarified by higher gauge theory [82,83]. But we may also hope for applications of higher gauge theory to other less speculative branches of physics, such as condensed matter physics.…”

Smooth functors vs. differential forms

“…The associated Lie 2-group, which also underlies the construction of a geometric realization of EG [Seg68] is here denoted by EG. It can be interpreted as the inner automorphism 2-group of G [RS08], and its Lie algebra plays an important role in [SSS09].…”